Simplified multiple input multiple output (mimo) communication schemes for interchip and intrachip communications

ABSTRACT

Aspects disclosed in the detailed description include simplified multiple input multiple output (MIMO) communication schemes for interchip and intrachip communications. In exemplary aspects, MIMO techniques are applied to interchip and intrachip communication systems. In particular, a programmable control function is provided at an electrical signal source and supports a default MIMO communication scheme on a MIMO channel comprising possible communication paths among a plurality of transmitting and receiving endpoints. In addition, the programmable control function can opportunistically employ a simplified MIMO communication scheme when the MIMO channel is determined to be a tri-diagonal MIMO channel. The simplified MIMO communication scheme uses an Inverse Fast Fourier Transformation (IFFT) with reduced computational complexity. By employing the simplified MIMO communication scheme, interchip or intrachip communication may be supported with reduced implementation complexity, lower power consumption, and improved robustness.

BACKGROUND

I. Field of the Disclosure

The technology of the disclosure relates generally to communicationtechniques between chips or between dies within a package.

II. Background

Computing devices have become common in modern society. The prevalenceof computing devices may be attributed to the many functions that areenabled within such computing devices. Increasingly complex integratedcircuits have been designed and manufactured to provide increasinglygreater functionality. Concurrent with the increases in complexity ofthe integrated circuits, there has been pressure to decrease the areaconsumed by the integrated circuits.

In many instances, the computing devices include a motherboard withseveral integrated circuits communicatively coupled to one anotherthrough conductive elements referred to as buses. Signals are passedfrom one integrated circuit to a second integrated circuit over suchbuses. As the complexity of the integrated circuits increases, thenumber of conductive elements required to convey signals between theintegrated circuits typically increases. Likewise, as the amount of dataincreases, the frequencies at which the data is transmitted increase. Asthe number of conductive elements increases and the frequencies alsoincrease, the opportunities for signals to interfere with one anotherincrease. This interference is commonly referred to as electromagneticinterference (EMI) or crosstalk. If the EMI is too severe, undesirableerrors may be introduced into the signal stream. While of concern forcommunication between two integrated circuits, EMI concerns also existfor communication that takes place between two dies within a singleintegrated circuit package.

Historically, each conductive element was treated as being functionallyindependent of other conductive elements even when the conductiveelements were proximate one another, such that crosstalk could occur.Because activity on one conductive element frequently impacts otherconductive elements, designers appreciate the ability to model theconductive elements more effectively so as to create more efficientcommunication schemes for communication between integrated circuits orbetween dies of a single integrated circuit package.

SUMMARY OF THE DISCLOSURE

Aspects disclosed in the detailed description include simplifiedmultiple input multiple output (MIMO) communication schemes forinterchip and intrachip communications. In exemplary aspects, MIMOtechniques are applied to interchip and intrachip communication systems.In particular, a programmable control function is provided at anelectrical signal source and supports a default MIMO communicationscheme on a MIMO channel comprising possible communication paths among aplurality of transmitting and receiving endpoints. In addition, theprogrammable control function can opportunistically employ a simplifiedMIMO communication scheme when the MIMO channel is determined to be atri-diagonal MIMO channel. The simplified MIMO communication scheme usesInverse Fast Fourier Transformation (IFFT) with reduced computationalcomplexity. By employing the simplified MIMO communication scheme,interchip or intrachip communication may be supported with reducedimplementation complexity, lower power consumption, and improvedrobustness.

In this regard in one aspect, an electrical signal source is provided.The electrical signal source comprises a plurality of communicationendpoints that are configured to transmit communication signals over aMIMO channel. The electrical signal source also comprises a programmablecontrol function communicatively coupled to the plurality ofcommunication endpoints. The programmable control function is configuredto receive a plurality of MIMO communication signals associated with afirst column vector. The programmable control function is furtherconfigured to determine if the MIMO channel is a tri-diagonal MIMOchannel. The programmable control function is further configured togenerate a second column vector that comprises the first column vectorand a plurality of add-on signal elements. The programmable controlfunction is further configured to provide the second column vector to anIFFT function to generate a plurality of transformed signals. Theprogrammable control function is further configured to cause a subset ofthe plurality of transformed signals to be transmitted over theplurality of communication endpoints.

In another aspect, an electrical signal source means is provided. Theelectrical signal source means comprises a means for communicationsignal transmission over a MIMO channel having a tri-diagonal channelresponse matrix. The electrical signal source means also comprises ameans for programmable control communicatively coupled to a plurality ofcommunication endpoints. The means for programmable control isconfigured to receive a plurality of MIMO communication signalsassociated with a first column vector. The means for programmablecontrol is also configured to generate a second column vector thatcomprises the first column vector and a plurality of add-on signalelements. The means for programmable control is also configured toprovide the second column vector to an IFFT function to generate aplurality of transformed signals. The means for programmable control isalso configured to cause a subset of the plurality of transformedsignals to be transmitted over the plurality of communication endpoints.

In another aspect, a method for transmitting communication signals overa MIMO channel having a tri-diagonal channel response matrix isprovided. The method comprises receiving a plurality of MIMOcommunication signals associated with a first column vector {d₁, d₂, . .. d_(N)}. The method also comprises generating a second column vector{d₀, d₁, d₂, . . . d_(N), d_(N+1), d_(N+2), . . . d_(2(N+1)−1)} thatcomprises the first column vector {d₁, d₂, . . . d_(N)} and a pluralityof add-on signal elements. The method also comprises providing thesecond column vector to an IFFT function to generate a plurality oftransformed signals associated with a third column vector {D₀, D₁, D₂, .. . D_(N), D_(N+1), D_(N+2), . . . D_(2(N+1)−1)}. The method alsocomprises providing a subset of the plurality of transformed signals{D₁, D₂, . . . D_(N)} from the third column vector to a plurality ofcommunication endpoints for transmission.

In another aspect, a MIMO communication system for interchip andintrachip communication is provided. The MIMO communication systemcomprises an electrical signal source. The electrical signal source alsocomprises a programmable control function communicatively coupled to afirst communication interface and a second communication interface. Theelectrical signal source also comprises a simplified MIMO functioncommunicatively coupled to the second communication interface and athird communication interface, wherein the simplified MIMO functioncomprises an IFFT function. The electrical signal source also comprisesa plurality of communication endpoints communicatively coupled to thethird communication interface. The MIMO communication system alsocomprises an electrical signal receiver. The electrical signal receivercomprises a plurality of receiver communication endpointscommunicatively coupled to a signal input interface. The electricalsignal receiver also comprises a computational function communicativelycoupled to the signal input interface and a signal output interface. Theelectrical signal receiver further comprises a receiver programmablecontrol function communicatively coupled to the signal output interface.The MIMO communication system also comprises a MIMO channel coupled withthe plurality of communication endpoints and the plurality of receivercommunication endpoints, wherein the MIMO channel is a tri-diagonal MIMOchannel.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a block diagram of an exemplary chip to chip communicationsystem that may benefit from exemplary aspects of the presentdisclosure;

FIG. 2A is a block diagram of an exemplary multiple input multipleoutput (MIMO) channel model of the chip to chip communication system ofFIG. 1;

FIG. 2B is a block diagram of an exemplary channel matrix of the MIMOchannel model of FIG. 2A;

FIG. 3A is a block diagram of an exemplary MIMO tri-diagonal channelmodel of a communication system;

FIG. 3B is a block diagram of an exemplary tri-diagonal channel matrixof the MIMO tri-diagonal channel model of FIG. 3A;

FIG. 4 is a block diagram of an exemplary MIMO-based electrical signalsource with a programmable control function configured to select adefault MIMO communication scheme or a simplified MIMO communicationscheme;

FIG. 5 is a flowchart of an exemplary MIMO operation control processused by the programmable control function of FIG. 4 to choose between adefault MIMO communication scheme and a simplified MIMO communicationscheme;

FIG. 6 is a block diagram of an exemplary MIMO-based electrical signalsource that implements a simplified MIMO communication scheme accordingto exemplary aspects of the present disclosure;

FIG. 7 is a block diagram of an exemplary electrical signal receiverthat implements the simplified MIMO communication scheme of FIG. 6according to exemplary aspects of the present disclosure;

FIG. 8 is a flowchart of an exemplary interchip and intrachipcommunication process for implementing the simplified MIMO communicationschemes of FIGS. 6 and 7; and

FIG. 9 is a block diagram of an exemplary processor-based system thatcan include the exemplary MIMO-based electrical signal source of FIG. 4.

DETAILED DESCRIPTION

With reference now to the drawing figures, several exemplary aspects ofthe present disclosure are described. The word “exemplary” is usedherein to mean “serving as an example, instance, or illustration.” Anyaspect described herein as “exemplary” is not necessarily to beconstrued as preferred or advantageous over other aspects.

Aspects disclosed in the detailed description include simplifiedmultiple input multiple output (MIMO) communication schemes forinterchip and intrachip communications. In exemplary aspects, MIMOtechniques are applied to interchip and intrachip communication systems.In particular, a programmable control function is provided at anelectrical signal source and supports a default MIMO communicationscheme on a MIMO channel comprising possible communication paths among aplurality of transmitting and receiving endpoints. In addition, theprogrammable control function can opportunistically employ a simplifiedMIMO communication scheme when the MIMO channel is determined to be atri-diagonal MIMO channel. The simplified MIMO communication scheme usesInverse Fast Fourier Transformation (IFFT) with reduced computationalcomplexity. By employing the simplified MIMO communication scheme,interchip or intrachip communication may be supported with reducedimplementation complexity, lower power consumption, and improvedrobustness.

Before discussing aspects of a MIMO-based electrical signal source thatincludes specific aspects of the present disclosure, a brief overview ofMIMO-based interchip communication systems that may incorporateexemplary aspects of the present disclosure is provided with referenceto FIGS. 1-3B. The discussion of specific exemplary aspects of aMIMO-based electrical signal source that comprises a programmablecontrol function begins with reference to FIG. 4.

In this regard, FIG. 1 is block diagram of an exemplary chip-to-chipcommunication system 10 that may benefit from exemplary aspects of thepresent disclosure. The interchip communication system 10 may include afirst integrated circuit (IC) or chip 12 (“Chip A”) and a second IC orchip 14 (“Chip B”). The chips 12, 14 may be positioned on a printedcircuit board (PCB) 16, such as through soldering or the like. The chips12, 14 are communicatively coupled by conductive elements 18 (sometimesreferred to as channels). The first chip 12 is coupled to the conductiveelements 18 by a plurality of endpoints 20(1)-20(N). Similarly, thesecond chip 14 is coupled to the conductive elements 18 by a pluralityof endpoints 22(1)-22(M). The notations “1-N” and “1-M” indicate thatany number of the referenced endpoints, 1-N and 1-M, respectively, maybe provided. It should be appreciated that the conductive elements 18may be routed between the first chip 12 and the second chip 14 byrouting software so as to minimize the distance traveled, while alsoproviding space for other elements (e.g., other chips, inductors,capacitors, or the like) on the PCB 16. The routing of the conductiveelements 18 is generally noted as a board layout 24. By treating theplurality of endpoints 20(1)-20(N) and the plurality of endpoints22(1)-22(M) as MIMO transmit and receive antennas, respectively, theinterchip communication system 10 may be supported as a MIMOcommunication system.

In this regard, FIG. 2A illustrates an exemplary MIMO channel model 10of the chip to chip communication system of FIG. 1. In FIG. 2A, thefirst chip 12 is a transmitter (Tx), and the second chip 14 is areceiver (Rx). Further, each of the plurality of endpoints 20(1)-20(N)is communicatively coupled to each of the plurality of endpoints22(1)-22(M) to varying degrees by virtue of electromagnetic interference(EMI) and crosstalk between the conductive elements 18. For example,endpoint 20(1) may be communicatively coupled to endpoints 22(1), 22(2),. . . 22(M), respectively. Likewise, endpoint 20(N) may also becommunicatively coupled to endpoints 22(1), 22(2), . . . 22(M),respectively. Thus, the coupling among the plurality of endpoints20(1)-20(N) and the plurality of endpoints 22(1)-22(M) may berepresented as an M×N channel matrix 26.

In this regard, FIG. 2B is a block diagram of an exemplary channelmatrix 28 of the MIMO channel model of FIG. 2A. Elements of FIG. 2A arereferenced in connection with FIG. 2B and will not be re-describedherein. According to FIG. 2B, the channel matrix 28 has M rows and Ncolumns, wherein M and N are finite integer numbers. In this regard, thechannel matrix 28 is referred to as an M×N channel matrix. Each of the Mrows corresponds to one of the plurality of endpoints 22(1)-22(M). Eachof the N columns corresponds to one of the plurality of endpoints20(1)-20(N). Accordingly, each of the elements in the channel matrix 28represents a communication signal transmitted between one of theplurality of endpoints 20(1)-20(N) in the first chip 12 and one of theplurality of endpoints 22(1)-22(M) in the second chip 14. For example,an element h_(i,j) (0≦i≦M, 0≦j≦N) in the channel matrix 28 represents acommunication signal transmitted from endpoint 20(j) in the first chip12 to endpoint 22(i) in the second chip 14.

With reference back to FIG. 2A, by treating the channels of theconductive elements 18 as interdependent instead of independent, MIMOsolutions may be applied to the chip-to-chip communication system 10 soas to form vectorized signaling using eigenvector beamforming at thefirst chip 12 and combining at the second chip 14 within the interchipcommunication system 10. Such MIMO solutions help eliminate or at leastmitigate the effects caused by crosstalk, reflections, limitedbandwidth, and jitter. In exemplary aspects of the present disclosure,orthogonal frequency division multiplexing (OFDM) may be used with theMIMO solutions so as to allow for frequency selective channels.

While the MIMO channel model of FIG. 2A and the channel matrix 28 ofFIG. 2B support all possible communication paths among the plurality ofendpoints 20(1)-20(N) and the plurality of endpoints 22(1)-22(M), notall of the possible communication paths are needed or utilized at alltimes. The MIMO channel model and the channel matrix 28 can besimplified when a subset of the communication paths is utilized. In thisregard, FIG. 3A illustrates an exemplary MIMO tri-diagonal channel modelof a communication system 10′. The communication system 10′ comprises atransmitter 12′ and a receiver 14′. In a non-limiting example, thetransmitter 12′ is a first chip and the receiver 14′ is a second chipwhen the communication system 10′ is provided for interchipcommunication. In another non-limiting example, the transmitter 12′ is afirst die and the receiver 14′ is a second die when the communicationsystem 10′ is provided for intrachip communication. The transmitter 12′comprises a plurality of endpoints 20′(1)-20′(N). The receiver 14′comprises a plurality of endpoints 22′(1)-22′(N). In a non-limitingexample, the transmitter 12′ and the receiver 14′ have an equal numberof N endpoints. Unlike in FIG. 2A, each of the plurality of endpoints20′(1)-20′(N) only communicates to adjacent endpoints from among theplurality of endpoints 22′(1)-22′(N). Similarly, each of the pluralityof endpoints 22′(1)-22′(N) only communicates to adjacent endpoints fromamong the plurality of endpoints 20′ (1)-20′(N). For example, endpoint20′(1) in the transmitter 12′ only communicates to endpoints 22′ (1) and22′ (2) in the receiver 14′. Endpoint 20′ (2) only communicates toendpoints 22′ (1), 22′ (2), and 22′ (3). Likewise, endpoint 22′ (1) inthe receiver 14′ only communicates to endpoints 20′(1) and 20′(2) in thetransmitter 12′. Endpoint 22′(2) only communicates to endpoints 20′ (1),20′ (2), and 20′ (3). A MIMO tri-diagonal channel model 26′ in thecommunication system 10′ can be represented by a N×N tri-diagonalchannel matrix, as discussed in more detail below.

In this regard, FIG. 3B is a block diagram of an exemplary tri-diagonalchannel matrix 28′ of the MIMO tri-diagonal channel model 26′ in FIG.3A. Elements of FIG. 3A are referenced in connection to FIG. 3B and willnot be re-described herein. According to FIG. 3B, a N row by N column(N×N) tri-diagonal channel matrix 28′ (represented by H₁) has non-zeroelements only on the main diagonal (e.g., the diagonal from elementh_(1,1) to element h_(N,N)) and the two adjacent diagonals (e.g., thediagonals above and below the main diagonal). All other elements in theN×N tri-diagonal channel matrix 28′ are zeroes. The tri-diagonal channelmatrix 28′ of FIG. 3B has special properties allowing a simplified MIMOcommunication scheme to be applied for interchip and intrachipcommunication.

In this regard, FIG. 4 is a block diagram of an exemplary MIMO-basedelectrical signal source 30 with a programmable control function 32configured to select a default MIMO communication scheme or a simplifiedMIMO communication scheme. The electrical signal source 30, which may bethe first chip 12 in FIG. 2A and/or the transmitter 12′ in FIG. 3A in anon-limiting example, comprises the programmable control function 32.The programmable control function 32, according to a non-limitingexample, may be implemented as a software-based function, or ahardware-based element, or a combination of both. The electrical signalsource 30 further comprises a default MIMO function 34 and a simplifiedMIMO function 36. The default MIMO function 34 is configured to supportthe MIMO channel model of FIG. 2A and the channel matrix 28 of FIG. 2B.In a non-limiting example, the default MIMO function 34 is configured tosupport the MIMO communication scheme as described in U.S. patentapplication Ser. No. 14/490,818, entitled “Multiple Input MultipleOutput (MIMO) Communication Systems and Methods for Chip to Chip andIntrachip Communication,” filed Sep. 19, 2014, and in U.S. ProvisionalPatent Application Ser. No. 62/032,027, entitled “Multiple InputMultiple Output (MIMO) Communication Systems and Methods for Chip toChip and Intrachip Communication,” filed Aug. 1, 2014, which are hereinincorporated by reference in their entireties. The simplified MIMOfunction 36 is configured to support the MIMO tri-diagonal channel modelof FIG. 3A and the tri-diagonal channel matrix 28′ of FIG. 3B. In anon-limiting example, the simplified MIMO function 36 implements asimplified MIMO communication scheme. More detail regarding thesimplified MIMO function 36 and the simplified MIMO communication schemeis provided with reference to FIGS. 6 and 7. The programmable controlfunction 32 is configured to toggle between the default MIMO function 34and the simplified MIMO function 36 according to a MIMO operationcontrol process discussed next.

In this regard, FIG. 5 is a flowchart of an exemplary MIMO operationcontrol process 38 used by the programmable control function 32 of FIG.4 to choose between a default MIMO communication scheme and a simplifiedMIMO communication scheme. Elements of FIG. 4 are referenced inconnection to FIG. 5 and will not be re-described herein. At the startof the MIMO operation control process 38, the programmable controlfunction 32 receives signals to be transmitted over a plurality ofendpoints (block 40). In response, the programmable control function 32determines whether a MIMO channel associated with the plurality ofendpoints is a tri-diagonal MIMO channel (block 42). If the MIMO channelassociated with the plurality of endpoints is a tri-diagonal MIMOchannel, the programmable control function 32 selects the simplifiedMIMO function 36 (block 44). Otherwise, the programmable controlfunction 32 selects the default MIMO function 34 (block 46). Aftermaking the selection, the programmable control function 32 sends thesignals to the selected MIMO function (block 48). The simplified MIMOcommunication scheme associated with the simplified MIMO function 36 isimplemented by a MIMO-based electrical signal source that is discussednext.

In this regard, FIG. 6 is a block diagram of an exemplary MIMO-basedelectrical signal source 50 that implements the simplified MIMOcommunication scheme according to exemplary aspects of the presentdisclosure. The electrical signal source 50 comprises a programmablecontrol function 52. The programmable control function 52 receives aplurality of communication signals 54(1)-54(N) over a firstcommunication interface 56. The plurality of communication signals54(1)-54(N) is associated with a first column vector {d₁, d₂, . . .d_(N)}, wherein each element represents one of the plurality ofcommunication signals 54(1)-54(N). For example, element d₁ representscommunication signal 54(1), element d₂ represents communication signal54(2), and so on such that element d_(N) represents communication signal54(N). In this regard, the first column vector contains N elementscorresponding to the N communication signals 54(1)-54(N), respectively.The programmable control function 52 in turn generates a second columnvector {d₀, d₁, d₂, . . . d_(N), d_(N+1), d_(N+2), . . . d_(2(N+1)−1)}that contains 2(N+1) elements. The second column vector contains all theelements of the first column vector {d₁, d₂, . . . d_(N)}. In addition,the second column vector includes N+2 add-on elements denoted by d₀,d_(N+1), d_(N+2), . . . d_(2(N+1)−1), respectively. All of the N+2add-on elements d₀, d_(N+1), d_(N+2), . . . d_(2(N+1)−1) have zerovalues. In other words, none of the N+2 add-on elements is associatedwith a real communication signal. The programmable control function 52provides the second column vector {d₀, d₁, d₂, . . . d_(N), d_(N+1),d_(N+2), . . . d_(2(N+1)−1)} to a simplified MIMO function 58 over asecond communication interface 60. An IFFT function 62, which works withthe simplified MIMO function 58 and transforms the second column vectorinto a plurality of transformed signals (not shown) associated with athird column vector {D₀, D₁, D₂, . . . D_(N), D_(N+1), D_(N+2), . . .D_(2(N+1)−1)}. A subset of transformed signals D₁, D₂, . . . D_(N) fromamong the plurality of transformed signals in the third column vector isof particular interest because they are the transformed counterparts ofthe plurality of communication signals 54(1)-54(N) associated with thefirst column vector {d₁, d₂, . . . d_(N)}. For example, D₁ is atransformed signal of d₁, D₂ is a transformed signal of d₂, and so on,such that D_(N) is a transformed signal of d_(N). Subsequently, thesimplified MIMO function 58 provides the subset of transformed signalsD₁, D₂, . . . D_(N) over a third communication interface 64 to aplurality of communication endpoints 66(1)-66(N) for transmission. In anon-limiting example, the plurality of communication endpoints66(1)-66(N) is equivalent to the plurality of endpoints 20′(1)-20′(N) inFIG. 3A. Although the programmable control function 52, the simplifiedMIMO function 58, and the IFFT function 62 are shown as separateelements in FIG. 6, there is nothing that prevents them from beingintegrated into a single element that is based on a software function, ahardware element, or a combination of both.

With continuing reference to FIG. 6, the subset of transformed signalsD₁, D₂, . . . D_(N) is transmitted from the plurality of communicationendpoints 66(1)-66(N) over the tri-diagonal MIMO channel (not shown) andreceived by an electrical signal receiver (not shown) also associatedwith the tri-diagonal MIMO channel (not shown). In this regard, FIG. 7is a block diagram of an exemplary electrical signal receiver 70 thatimplements the simplified MIMO communication scheme of FIG. 6 accordingto exemplary aspects of the present disclosure. Elements of FIG. 6 arereferenced in connection to FIG. 7 and will not be re-described herein.The electrical signal receiver 70 comprises a plurality of receivercommunication endpoints 72(1)-72(N) that are communicatively coupled tothe plurality of communication endpoints 66(1)-66(N) over thetri-diagonal MIMO channel (not shown). In a non-limiting example, theplurality of receiver communication endpoints 72(1)-72(N) is equivalentto the plurality of endpoints 22′(1)-22′(N) in FIG. 3A. The subset oftransformed signals D₁, D₂, . . . D_(N) is received by the plurality ofreceiver communication endpoints 72(1)-72(N) over the tri-diagonal MIMOchannel (not shown) and provided to a computational function 74 in theelectrical signal receiver 70 over a signal input interface 76. In anon-limiting example, the computational function 74 may be implementedas a software function, a hardware element, or a combination of both.Due to transformations performed by the IFFT function 62 in theelectrical signal source 50, each of the subset of transformed signalsD₁, D₂, . . . D_(N) is a complex conjugation comprising a real part andan imaginary part. The computational function 74 takes the imaginarypart of the subset of transformed signals D₁, D₂, . . . D_(N) andgenerates a plurality of received signals X₁, X₂, . . . X_(N) associatedwith a fourth column vector {X₁, X₂, . . . X_(N)}. The fourth columnvector {X₁, X₂, . . . X_(N)} is provided to a receiver programmablecontrol function 78 over a signal output interface 80. The programmablecontrol function 78, according to a non-limiting example, may beimplemented as a software-based function, a hardware-based element, or acombination of both. Although the computational element 74 and thereceiver programmable control function 78 are shown as separate elementsin FIG. 7, there is nothing that prevents them from being integratedinto a single element that is based on a software function, a hardwareelement, or a combination of both.

To ascertain the received signals X₁, X₂, . . . X_(N) produced by thecomputational function 74, evaluations of a unitary transformation areprovided next. When the IFFT function 62 is not applied in theelectrical signal source 50, the unitary transformation is a matrixmultiplication that may be expressed as the following equation (Eq. 1):

x=Hd

wherein:

d is a N×1 column vector {d₁, d₂, . . . d_(N)} representing Ntransmitted signals (e.g., the first column vector in FIG. 6);

x is a N×1 column vector {X₁, X₂, . . . X_(N)} representing N receivedsignals (e.g., the fourth column vector in FIG. 7); and

H is a N×N unitary matrix with the property UU*=U*U=I, wherein I is anN×N identity matrix that has a value one (1) on the main diagonal and avalue zero (0) elsewhere in the matrix.

For the N×N unitary martix H of the following form,

$H{\text{:}\mspace{14mu}\begin{bmatrix}{\sin \left( \frac{\pi}{N + 1} \right)} & {\sin \left( \frac{2\; \pi}{N + 1} \right)} & {\sin \left( \frac{3\; \pi}{N + 1} \right)} & {\sin \left( \frac{4\; \pi}{N + 1} \right)} & \ldots & {\sin \left( \frac{\left( {N - 1} \right)\pi}{N + 1} \right)} & {\sin \left( \frac{N\; \pi}{N + 1} \right)} \\{\sin \left( \frac{{1 \cdot 2}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{2 \cdot 2}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{3 \cdot 2}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{4 \cdot 2}\; \pi}{N + 1} \right)} & \ldots & {\sin \left( \frac{{\left( {N - 1} \right) \cdot 2}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{N \cdot 2}\; \pi}{N + 1} \right)} \\{\sin \left( \frac{{1 \cdot 3}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{2 \cdot 3}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{3 \cdot 3}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{4 \cdot 3}\; \pi}{N + 1} \right)} & \ldots & {\sin \left( \frac{{\left( {N - 1} \right) \cdot 3}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{N \cdot 3}\; \pi}{N + 1} \right)} \\\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\{\sin \left( \frac{{1 \cdot \left( {N - 2} \right)}\pi}{N + 1} \right)} & {\sin \left( \frac{{2 \cdot \left( {N - 2} \right)}\pi}{N + 1} \right)} & {\sin \left( \frac{{3 \cdot \left( {N - 2} \right)}\pi}{N + 1} \right)} & \ldots & {\sin \left( \frac{\left( {N - 2} \right)\left( {N - 2} \right)\pi}{N + 1} \right)} & {\sin \left( \frac{\left( {N - 1} \right)\left( {N - 2} \right)\pi}{N + 1} \right)} & {\sin \left( \frac{{N\left( {N - 2} \right)}\pi}{N + 1} \right)} \\{\sin \left( \frac{{1 \cdot \left( {N - 1} \right)}\pi}{N + 1} \right)} & {\sin \left( \frac{{2 \cdot \left( {N - 1} \right)}\pi}{N + 1} \right)} & {\sin \left( \frac{{3 \cdot \left( {N - 1} \right)}\pi}{N + 1} \right)} & \ldots & {\sin \left( \frac{\left( {N - 2} \right)\left( {N - 1} \right)\pi}{N + 1} \right)} & {\sin \left( \frac{\left( {N - 1} \right)\left( {N - 1} \right)\pi}{N + 1} \right)} & {\sin \left( \frac{{N\left( {N - 1} \right)} \cdot \pi}{N + 1} \right)} \\{\sin \left( \frac{{1 \cdot N}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{2 \cdot N}\; \pi}{N + 1} \right)} & {\sin \left( \frac{{3 \cdot N}\; \pi}{N + 1} \right)} & \ldots & {\sin \left( \frac{\left( {N - 2} \right)N\; \pi}{N + 1} \right)} & {\sin \left( \frac{\left( {N - 1} \right)N\; \pi}{N + 1} \right)} & {\sin \left( \frac{{N \cdot N}\; \pi}{N + 1} \right)}\end{bmatrix}}$

matrix multiplication according to Eq. 1 produces an x column vector{X₁, X₂, . . . X_(N)} that can be expressed by the following equations(Eq. 2):

$X_{1} = {{{d_{1}{\sin \left( \frac{\pi}{N + 1} \right)}} + {d_{2}{\sin \left( \frac{2\; \pi}{N + 1} \right)}} + {d_{3}{\sin \left( \frac{3\; \pi}{N + 1} \right)}} + \ldots + {d_{N - 1}{\sin \left( \frac{\left( {N - 1} \right)\pi}{N + 1} \right)}} + {d_{N}{\sin \left( \frac{N\; \pi}{N + 1} \right)}}} = {\sum\limits_{n = 1}^{N}{d_{n}{\sin \left( \frac{\pi \cdot n}{N + 1} \right)}}}}$$X_{2} = {{{d_{1}{\sin \left( \frac{2\; \pi}{N + 1} \right)}} + {d_{2}{\sin \left( \frac{{2 \cdot 2}\; \pi}{N + 1} \right)}} + {d_{3}{\sin \left( \frac{{3 \cdot 2}\; \pi}{N + 1} \right)}} + \ldots + {d_{N - 1}{\sin \left( \frac{{\left( {N - 1} \right) \cdot 2}\; \pi}{N + 1} \right)}} + {d_{N}{\sin \left( \frac{{N \cdot 2}\; \pi}{N + 1} \right)}}} = {\sum\limits_{n = 1}^{N}{d_{n}{\sin \left( \frac{2\; {\pi \cdot n}}{N + 1} \right)}}}}$     ⋮$X_{N - 1} = {{{d_{1}{\sin \left( \frac{{1 \cdot \left( {N - 1} \right)}\pi}{N + 1} \right)}} + {d_{2}{\sin \left( \frac{{2 \cdot \left( {N - 1} \right)}\pi}{N + 1} \right)}} + {d_{3\;}{\sin \left( \frac{{3 \cdot \left( {N - 1} \right)}\pi}{N + 1} \right)}} + {\ldots \mspace{14mu} d_{N}{\sin \left( \frac{{N \cdot \left( {N - 1} \right)}\pi}{N + 1} \right)}}} = {\sum\limits_{n = 1}^{N}{d_{n}{\sin \left( \frac{\left( {N - 1} \right){\pi \cdot n}}{N + 1} \right)}}}}$$X_{N} = {{{d_{1}{\sin \left( \frac{{1 \cdot N}\; \pi}{N + 1} \right)}} + {d_{2}{\sin \left( \frac{{2 \cdot N}\; \pi}{N + 1} \right)}} + {d_{3}{\sin \left( \frac{{3 \cdot N}\; \pi}{N + 1} \right)}} + \ldots + {d_{N - 1}{\sin \left( \frac{\left( {N - 1} \right)N\; \pi}{N + 1} \right)}} + {d_{N}{\sin \left( \frac{N^{2}\pi}{N + 1} \right)}}} = {\sum\limits_{n = 1}^{N}{d_{n}{\sin \left( \frac{N\; {\pi \cdot n}}{N + 1} \right)}}}}$

As previously described in reference to FIG. 6, an IFFT performed by theIFFT function 62 on the second column vector {d₀, d₁, d₂, . . . d_(N),d_(N+1), d_(N+2), . . . d_(2(N+1)−1)} generates the plurality oftransformed signals represented by the third column vector {D₀, D₁, D₂,. . . D_(N), D_(N+1), D_(N+2), . . . D_(2(N+1)−1)}. The subset oftransformed signals D₁, D₂, . . . D_(N), which is actually transmittedby the plurality of communication endpoints 66(1)-66(N), may beexpressed by the following equations (Eq. 3):

$D_{1} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot n}}{K} \right)}}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot n}}{2\left( {N + 1} \right)} \right)}}} = {\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; {\pi \cdot n}}{\left( {N + 1} \right)} \right)}}}}}$$D_{2} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot 2 \cdot n}}{K} \right)}}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot 2 \cdot n}}{2\left( {N + 1} \right)} \right)}}} = {\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; {\pi \cdot 2 \cdot n}}{\left( {N + 1} \right)} \right)}}}}}$$D_{3} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot 3 \cdot n}}{K} \right)}}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot 3 \cdot n}}{2\left( {N + 1} \right)} \right)}}} = {\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; {\pi \cdot 3 \cdot n}}{\left( {N + 1} \right)} \right)}}}}}$⋮$D_{N} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot N \cdot n}}{K} \right)}}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; 2\; {\pi \cdot N \cdot n}}{2\left( {N + 1} \right)} \right)}}} = {\sum\limits_{n = 1}^{n = N}{d_{n}{\exp \left( \frac{j\; {\pi \cdot N \cdot n}}{\left( {N + 1} \right)} \right)}}}}}$

Further, as described in reference to FIG. 7, the computational function74 takes the imaginary part of the subset of transformed signals D₁, D₂,. . . D_(N) and generates the plurality of received signals associatedwith the fourth column vector {X₁, X₂, . . . X_(N)}. Elements of thefourth column vector are expressed by the following equations (Eq. 4):

${{Im}\left\{ D_{1} \right\}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\sin \left( \frac{j\; {\pi \cdot n}}{\left( {N + 1} \right)} \right)}}} = X_{1}}$${{Im}\left\{ D_{2} \right\}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\sin \left( \frac{j\; {\pi \cdot 2 \cdot n}}{\left( {N + 1} \right)} \right)}}} = X_{2}}$${{Im}\left\{ D_{3} \right\}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\sin \left( \frac{j\; {\pi \cdot 3 \cdot n}}{\left( {N + 1} \right)} \right)}}} = X_{2}}$⋮${{Im}\left\{ D_{N} \right\}} = {{\sum\limits_{n = 1}^{n = N}{d_{n}{\sin \left( \frac{j\; {\pi \cdot N \cdot n}}{\left( {N + 1} \right)} \right)}}} = X_{N}}$

Not coincidentally, the x column vector produced by matrixmultiplication as in Eq. 2 is identical to the fourth column vectorproduced by IFFT unitary transformations as in Eq. 3 and Eq. 4, thus,validating the simplified MIMO communication scheme according toexemplary aspects of the present disclosure.

In this regard, FIG. 8 is a flowchart of an exemplary interchip andintrachip communication process 90 for implementing the simplified MIMOcommunication schemes in FIGS. 6 and 7. Elements of FIG. 4 arereferenced in connection to FIG. 8 and will not be re-described herein.According to the interchip and intrachip communication process 90, atransmitter (TX) programmable control function 32 first receives aplurality of MIMO communication signals associated with a first columnvector {d₁, d₂, . . . d_(N)} (block 92). The TX programmable controlfunction 32 then generates a second column vector {d₀, d₁, d₂, . . .d_(N), d_(N+1), d_(N+2), . . . d_(2(N+1)−1)} that comprises the firstcolumn vector {d₁, d₂, . . . d_(N)} and a plurality of add-on signalelements (block 94). The TX programmable control function 32 thenassigns zero values to each of the plurality of add-on signal elementsin the second column vector (block 96). As a result, the second columnvector has a first element d₀ as zero (0) value, followed by elementsd₁, d₂, . . . d_(N) from the first column vector, and followed byremaining elements d_(N+1), d_(N+2), . . . d_(2(N+1)−1) all having zero(0) values. Next, the TX programmable control function 32 provides thesecond column vector to an IFFT function 62 to generate a plurality oftransformed signals {D₀, D₁, D₂, . . . D_(N), D_(N+1), D_(N+2), . . .D_(2(N+1)−1)} (block 98). Then, the TX programmable control function 32provides the transformed signals {D₁, D₂, . . . D_(N)} to communicationendpoints 66(1)-66(N) for transmission (block 100). The remainingtransformed signals D₀, D_(N+1), D_(N+2), . . . D_(2(N+1)−1) arediscarded. Upon receiving the transformed signals {D₁, D₂, . . . D_(N)},a receiver (RX) programmable control function takes imaginary parts ofthe transformed signals {D₁, D₂, . . . D_(N)} to generate the receivedsignals {X₁, X₂, . . . X_(N)} (block 102).

Simplified MIMO communication schemes for interchip and intrachipcommunications according to aspects disclosed herein may be provided inor integrated into any processor-based device. Examples, withoutlimitation, include a set top box, an entertainment unit, a navigationdevice, a communications device, a fixed location data unit, a mobilelocation data unit, a mobile phone, a cellular phone, a computer, aportable computer, a desktop computer, a personal digital assistant(PDA), a monitor, a computer monitor, a television, a tuner, a radio, asatellite radio, a music player, a digital music player, a portablemusic player, a digital video player, a video player, a digital videodisc (DVD) player, and a portable digital video player.

In this regard, FIG. 9 illustrates an example of a processor-basedsystem 104 that can employ the simplified MIMO communication schemeillustrated in FIGS. 4-8. In this example, the processor-based system104 includes one or more central processing units (CPUs) 106, eachincluding one or more processors 108. The CPU(s) 106 may have cachememory 110 coupled to the processor(s) 108 for rapid access totemporarily stored data. The CPU(s) 106 is coupled to a system bus 112and can intercouple devices included in the processor-based system 104.As is well known, the CPU(s) 106 communicates with these other devicesby exchanging address, control, and data information over the system bus112. Although not illustrated in FIG. 9, multiple system buses 112 couldbe provided, wherein each system bus 112 constitutes a different fabric.

Other devices can be connected to the system bus 112. As illustrated inFIG. 9, these devices can include a memory system 114, one or more inputdevices 116, one or more output devices 118, one or more networkinterface devices 120, and one or more display controllers 122, asexamples. The input device(s) 116 can include any type of input device,including but not limited to input keys, switches, voice processors,etc. The output device(s) 118 can include any type of output device,including but not limited to audio, video, other visual indicators, etc.The network interface device(s) 120 can be any devices configured toallow exchange of data to and from a network 124. The network 124 can beany type of network, including but not limited to a wired or wirelessnetwork, a private or public network, a local area network (LAN), awireless local area network (WLAN), a wireless wide area network (WWAN),and the Internet. The network interface device(s) 120 can be configuredto support any type of communications protocol desired. The memorysystem 114 can include one or more memory units (not shown).

The CPU(s) 106 may also be configured to access the displaycontroller(s) 122 over the system bus 112 to control information sent toone or more displays 126. The display controller(s) 122 sendsinformation to the display(s) 126 to be displayed via one or more videoprocessors 128, which process the information to be displayed into aformat suitable for the display(s) 126. The display(s) 126 can includeany type of display, including but not limited to a cathode ray tube(CRT), a LED display, a liquid crystal display (LCD), a plasma display,etc.

Those of skill in the art will further appreciate that the variousillustrative logical blocks, modules, circuits, and algorithms describedin connection with the aspects disclosed herein may be implemented aselectronic hardware, instructions stored in memory or in anothercomputer-readable medium and executed by a processor or other processingdevice, or combinations of both. The master devices, and slave devicesdescribed herein may be employed in any circuit, hardware component,integrated circuit (IC), or IC chip, as examples. Memory disclosedherein may be any type and size of memory and may be configured to storeany type of information desired. To clearly illustrate thisinterchangeability, various illustrative components, blocks, modules,circuits, and steps have been described above generally in terms oftheir functionality. How such functionality is implemented depends uponthe particular application, design choices, and/or design constraintsimposed on the overall system. Skilled artisans may implement thedescribed functionality in varying ways for each particular application,but such implementation decisions should not be interpreted as causing adeparture from the scope of the present disclosure.

The various illustrative logical blocks, modules, and circuits describedin connection with the aspects disclosed herein may be implemented orperformed with a processor, a Digital Signal Processor (DSP), anApplication Specific Integrated Circuit (ASIC), a Field ProgrammableGate Array (FPGA) or other programmable logic device, discrete gate ortransistor logic, discrete hardware components, or any combinationthereof designed to perform the functions described herein. A processormay be a microprocessor, but in the alternative, the processor may beany conventional processor, controller, microcontroller, or statemachine. A processor may also be implemented as a combination ofcomputing devices, e.g., a combination of a DSP and a microprocessor, aplurality of microprocessors, one or more microprocessors in conjunctionwith a DSP core, or any other such configuration.

The aspects disclosed herein may be embodied in hardware and ininstructions that are stored in hardware, and may reside, for example,in Random Access Memory (RAM), flash memory, Read Only Memory (ROM),Electrically Programmable ROM (EPROM), Electrically ErasableProgrammable ROM (EEPROM), registers, a hard disk, a removable disk, aCD-ROM, or any other form of computer readable medium known in the art.An exemplary storage medium is coupled to the processor such that theprocessor can read information from, and write information to, thestorage medium. In the alternative, the storage medium may be integralto the processor. The processor and the storage medium may reside in anASIC. The ASIC may reside in a remote station. In the alternative, theprocessor and the storage medium may reside as discrete components in aremote station, base station, or server.

It is also noted that the operational steps described in any of theexemplary aspects herein are described to provide examples anddiscussion. The operations described may be performed in numerousdifferent sequences other than the illustrated sequences. Furthermore,operations described in a single operational step may actually beperformed in a number of different steps. Additionally, one or moreoperational steps discussed in the exemplary aspects may be combined. Itis to be understood that the operational steps illustrated in the flowchart diagrams may be subject to numerous different modifications aswill be readily apparent to one of skill in the art. Those of skill inthe art will also understand that information and signals may berepresented using any of a variety of different technologies andtechniques. For example, data, instructions, commands, information,signals, bits, symbols, and chips that may be referenced throughout theabove description may be represented by voltages, currents,electromagnetic waves, magnetic fields or particles, optical fields orparticles, or any combination thereof.

The previous description of the disclosure is provided to enable anyperson skilled in the art to make or use the disclosure. Variousmodifications to the disclosure will be readily apparent to thoseskilled in the art, and the generic principles defined herein may beapplied to other variations without departing from the spirit or scopeof the disclosure. Thus, the disclosure is not intended to be limited tothe examples and designs described herein, but is to be accorded thewidest scope consistent with the principles and novel features disclosedherein.

What is claimed is:
 1. An electrical signal source, comprising: aplurality of communication endpoints configured to transmitcommunication signals over a multiple input multiple output (MIMO)channel; a programmable control function communicatively coupled to theplurality of communication endpoints, the programmable control functionconfigured to receive a plurality of MIMO communication signalsassociated with a first column vector; and wherein the programmablecontrol function, further configured to: determine if the MIMO channelis a tri-diagonal MIMO channel; generate a second column vectorcomprising: the first column vector; and a plurality of add-on signalelements; provide the second column vector to an Inverse Fast FourierTransformation (IFFT) function to generate a plurality of transformedsignals; and cause a subset of the plurality of transformed signals tobe transmitted over the plurality of communication endpoints.
 2. Theelectrical signal source of claim 1, wherein the first column vector is{d₁, d₂, . . . d_(N)}.
 3. The electrical signal source of claim 2,wherein the plurality of add-on signal elements are denoted by d₀,d_(N+1), d_(N+2), . . . d_(2(N+1)−1), each of the plurality of add-onsignal elements is set to zero (0).
 4. The electrical signal source ofclaim 3, wherein the second column vector is {d₀, d₁, d₂, . . . d_(N),d_(N+1), d_(N+2), . . . d_(2(N+1)−1)}.
 5. The electrical signal sourceof claim 4, wherein the plurality of transformed signals is associatedwith a third column vector {D₀, D₁, D₂, . . . D_(N), D_(N+1), D_(N+2), .. . D_(2(N+1)−1)}.
 6. The electrical signal source of claim 5, whereinthe subset of the plurality of transformed signals to be transmittedover the plurality of communication endpoints is associated withelements D₁, D₂, . . . D_(N) from the third column vector.
 7. Theelectrical signal source of claim 1, wherein the programmable controlfunction is configured to invoke a default MIMO communication method ifthe MIMO channel is determined to be a non-tri-diagonal MIMO channel. 8.The electrical signal source of claim 1 integrated into an integratedcircuit (IC).
 9. The electrical signal source of claim 1 integrated intoa device selected from the group consisting of: a set top box; anentertainment unit; a navigation device; a communications device; afixed location data unit; a mobile location data unit; a mobile phone; acellular phone; a computer; a portable computer; a desktop computer; apersonal digital assistant (PDA); a monitor; a computer monitor; atelevision; a tuner; a radio; a satellite radio; a music player; adigital music player; a portable music player; a digital video player; avideo player; a digital video disc (DVD) player; and a portable digitalvideo player.
 10. An electrical signal source means, comprising: a meansfor communication signal transmission over a multiple input multipleoutput (MIMO) channel having a tri-diagonal channel response matrix; ameans for programmable control communicatively coupled to a plurality ofcommunication endpoints, the means for programmable control configuredto receive a plurality of MIMO communication signals associated with afirst column vector; and wherein the means for programmable control,further configured to: generate a second column vector comprising: thefirst column vector; and a plurality of add-on signal elements; providethe second column vector to an Inverse Fast Fourier Transformation(IFFT) function to generate a plurality of transformed signals; andcause a subset of the plurality of transformed signals to be transmittedover the plurality of communication endpoints.
 11. A method fortransmitting communication signals over a multiple input multiple output(MIMO) channel having a tri-diagonal channel response matrix,comprising: receiving a plurality of MIMO communication signalsassociated with a first column vector {d₁, d₂, . . . d_(N)}; generatinga second column vector {d₀, d₁, d₂, . . . d_(N), d_(N+1), d_(N+2), . . .d_(2(N+1)−1)}, the second column vector comprises the first columnvector {d₁, d₂, . . . d_(N)} and a plurality of add-on signal elements;providing the second column vector to an Inverse Fast FourierTransformation (IFFT) function to generate a plurality of transformedsignals associated with a third column vector {D₀, D₁, D₂, . . . D_(N),D_(N+1), D_(N+2), . . . D_(2(N+1)−1)}; and providing a subset oftransformed signals {D₁, D₂, . . . D_(N)} from the third column vectorto a plurality of communication endpoints for transmission.
 12. Themethod of claim 11, further comprising assigning zero value to each ofthe plurality of add-on signal elements.
 13. The method of claim 11,further comprising generating received signals associated with a fourthcolumn vector {X₁, X₂, . . . X_(N)} by taking imaginary parts of thesubset of transformed signals {D₁, D₂, . . . D_(N)}.
 14. A multipleinput multiple output (MIMO) communication system for communication,comprising: an electrical signal source, comprising: a programmablecontrol function communicatively coupled to a first communicationinterface and a second communication interface; a simplified MIMOfunction communicatively coupled to the second communication interfaceand a third communication interface, the simplified MIMO functioncomprises an Inverse Fast Fourier Transformation (IFFT) function; and aplurality of communication endpoints communicatively coupled to thethird communication interface; an electrical signal receiver,comprising: a plurality of receiver communication endpointscommunicatively coupled to a signal input interface; a computationalfunction communicatively coupled to the signal input interface and asignal output interface; and a receiver programmable control functioncommunicatively coupled to the signal output interface; and a MIMOchannel coupled with the plurality of communication endpoints and theplurality of receiver communication endpoints, wherein the MIMO channelis a tri-diagonal MIMO channel.
 15. The MIMO communication system ofclaim 14, wherein the programmable control function is a software-basedfunction.
 16. The MIMO communication system of claim 14, wherein theprogrammable control function is a hardware-based element.
 17. The MIMOcommunication system of claim 14, wherein the simplified MIMO functionis a software-based function.
 18. The MIMO communication system of claim14, wherein the simplified MIMO function is a hardware-based element.19. The MIMO communication system of claim 14, wherein the simplifiedMIMO function is integrated with the programmable control function. 20.The MIMO communication system of claim 14, wherein the receiverprogrammable control function is a software-based function.
 21. The MIMOcommunication system of claim 14, wherein the receiver programmablecontrol function is a hardware-based element.
 22. The MIMO communicationsystem of claim 14, wherein the computational function is configured to:receive a plurality of MIMO communication signals on the signal inputinterface; extract an imaginary part of the plurality of MIMOcommunication signals; and provide the imaginary part of the pluralityof MIMO communication signals to the signal output interface.
 23. TheMIMO communication system of claim 14, wherein the computationalfunction is a software-based function.
 24. The MIMO communication systemof claim 14, wherein the computational function is a hardware-basedelement.
 25. The MIMO communication system of claim 14, wherein thecomputational function is integrated with the receiver programmablecontrol function.